Convective heat and mass transfer rate on 3D Williamson nanofluid flow via linear stretching sheet with thermal radiation and heat absorption

A mathematical analysis is communicated to the thermal radiation and heat absorption effects on 3D MHD Williamson nanoliquid (NFs) motion via stretching sheet. The convective heat and mass boundary conditions are taken in sheet when liquid is motion. As a novelty, the effects of thermal radiation, heat absorption and heat and mass convection are incorporated. The aim is to develop heat transfer. Williamson NFs are most important source of heat absorption, it having many significant applications in “energy generation, HT, aircraft, missiles, electronic cooling systems, gas turbines” etc. The suitable similarity transformations have been utilized for reduce basic governing P.D. E’s into coupled nonlinear system of O.D. E’s. Obtained O.D. Es are calculated by help of R–K–F (“Runge–Kutta–Fehlberg”)4th order procedure with shooting technique in MATLAB programming. We noticed that, the skin friction coefficient is more effective in Williamson liquid motion when compared with NFs motion with higher numerical values of stretching ratio parameter, Williamson liquid motion is high when compared to NFs motion for large values of magnetic field. We compared with present results into previous results for various conditions. Finally, in the present result is good invention of previous results.


Abbreviations
The MHD reflects dynamic activities via a stretching surface (SS). This liquid motion is electrically conducting, it established with magnetic field. The electrically conducting and heat transfer (HT) motion has many applications ("configuration orientation regarding structure of boundary layer, energy extractions in geothermal field, MHD accelerators and power generators, fluid droplets sprays and flow meters, electrostatic precipitation, polymer technology, centrifugal separation of matter from fluid, petroleum industry, magnetohydrodynamic generators, cooling systems, metallurgy, nuclear reactors, crystal growth, fluid metals and aerodynamics, accelerators, pumps, solar physics, plasma confinement, cosmology ext. ") in modern industry and engineering fields, science technology. The introduction of Magnetohydrodynamics (MHD) was established by Roberts 32 . Jamil and Haleem 33 obtained unsteady motion of fractionalized magnetohydrodynamic Jeffrey liquid via porous plate with linear slip effect. The natural convection motion of non-Newtonian liquid via different surface was focused [34][35][36] . The convection motion via various sheets was explored [37][38][39] . The MHD motion of Erying-Powell liquid via SS was discussed [40][41][42] . The clearance between ceramic outer ring and steel pedestal on sound radiation was discussed 43 . Some of authors [44][45][46] presented Eyring-Powell liquid via SS. He et al. 47 presented microwave imaging of 3D dielectric-magnetic penetrable objects. Tamoor et al. 48 exhibited the MHD Casson liquid motion induced by stretched cylinder. MHD peristaltic transport of NFs ("copper-water") in artery with mild stenosis for different shapes of nanoparticles is studied Devaki et al. 49 . Mahabaleshwar et al. 50 examined the MHD Couple stress liquid due to perforated sheet via linear stretching with radiation. The MHD motion via SS was presented by [51][52][53] . Recently, some of investigators 54-56 developed suction or injection for gravity modulation mixed convection in micropolar liquid via inclined sheet. The 3D MHD non-Newtonian NFs via SS was exhibited [57][58][59][60] . Some of the interesting and related research was studied by [61][62][63] . Recently [64][65][66] , developed MHD and convective heat transfer motion via SS. The non-Newtonian liquid motion was studied [67][68][69] .
The impact of thermal radiative (TR) motion has normally known as variance b/w ambient energy and thermal energy. Which is used in several fields ("biomedicine, space machinery, drilling process, cancer treatment, high temperature methods, and power generation etc. Also, several industrial processes, include nuclear reactors, power plants, gas turbines, satellites, missiles technology etc. ") of technologies. Satya Narayana et al. 70 focussed the thermal radiation effect on unsteady motion via SS. Kandasamy et al. 71 analysed thermal and Solutal effect on heat ad mass transfer induced due to a NFs via porous vertical plate. The mixed convection and TR on nonaligned Casson liquid via SS was discussed Mehmood et al. 72 exhibited the radiative motion on 2D Casson liquid past a moving wedge. Masthanaiah et al. 73 presented heat generation on cold liquid with viscous dissipation via parallel plates. The NFs motion via radiative sheet was examined 74,75 . Recently, some of related, interacted and motivated work presented [76][77][78] .
The current work numerical analysis, which is enables the young researchers to compute of convective heat and mass transfer on 3D Williamson NFs motion via SS. The effect of thermal radiation, heat absorption, MHD, convective heat and mass transfer are considered in this study. It has several applications in industrial processes, petroleum industry, nuclear reactors, power plants, molecular weight polymers, energy extractions in geothermal field, power generators, polymer technology, magnetohydrodynamic generators, cooling systems, nuclear reactors, crystal growth, biomedicine, space machinery, drilling process, cancer treatment, etc.
The main motivations of current work are: (a) The convective heat and mass transfer boundary conditions on Williamson NFs motion, (b). The effect of heat absorption and thermal radiation is enhancement of heat transportation in Williamson NFs via SS.
(c) Particularly, numerical values of "Thermal Radiation", "Magnetic field", "Lewis number and Prandtl numbers" leads to minimum heat and mass transfer rate are obtained.
The present results are justified through comparison by previous study as shown Table 1 and Table 2. Table 1. Comparison of Initial values in the absence of = 0 and α = 0.
α Mathematical formulation. Convective heat and mass transfer on 3D magnetohydrodynamic Williamson nanoliquid motion via linear stretching surface with chemical reaction is consider. Which is assumed that stretching along x * , y * -surface, the fluid flow direction along z * > 0 and flow is induced by a stretching at z * = 0 as displayed in Fig. 1. The non-uniform magnetic field M 0 is taken in liquid motion direction. The stretching velocities along x * , y * -directions as U * w = a 1 x * and V * w = b 1 y * is considered, respectively. The general equations of the Williamson liquid motion for conservative of mass, conservative of momentum is given below: Williamson liquid model Equations are given by Ref. 12,15,81,82 : The "first Rivlin-Erickson tensor" A 1 and shear rate γ * is defined as below: www.nature.com/scientificreports/ Consider µ * ∞ = 0 and Ŵ * γ * < 1 thus Eq. (4) can be expressed as Under consideration of above, the governing equations of conservative of mass, conservative of momentum, conservative of energy and concentration are formed as following Ref. 12 : The boundary conditions of the present model are The radiative heat flux q r which is given by Quinn Brewster 83 is given by Neglected higher order terms we get Differentiate above heat flux equation, we get Substituting Eq. (14) in Eq. (4), we get below Expression The similarity transformations as beloẇ as z * = 0 as
The numerical results on f ′′ (0) ("Velocity Gradients") as = 0 for various values of α in Table 1. Also, Table 2 exhibited coefficient of skin friction with different results of α for = 0 . The outcomes are matched with those of Wang 79 , Ariel et al. 80 . It is noticed that very good agreement up to eight decimal places. Tables 3 and 4 Explored the Heat and Mass Transfer rates with various numerical numbers for α = 0.

Conclusions
This article related to the influence of thermal radiative and heat absorption on 3D Williamson nanoliquid motion via linear stretching surface. It is analyzed numerical technique with 4th order R-K-F ("Runge-Kutta-Fehlberg") scheme. We have noticed, the main points in present mathematical model as below: • The velocity of Williamson nanofluid motion is high when compared to nanofluid motion with high effect of M. • Skinfriction coefficient ("along x * -axis") is high in absents of α("Williamson parameter") while comparing to presence of α("Williamson parameter") with higher statistical values of . • Skinfriction coefficient ("along x * -axis") is high in Williamson liquid motion while comparing to nanoliquid motion with higher statistical values of α. • The temperature is high in non-newtonian nanofluid while compared with enhance statistical values.  www.nature.com/scientificreports/